We initiate the study of the Interval Selection problem in the (streaming) sliding window model of computation. In this problem, an algorithm receives a potentially infinite stream of intervals on the line, and the objective is to maintain at every moment an approximation to a largest possible subset of disjoint intervals among the $L$ most recent intervals, for some integer $L$. We give the following results: - In the unit-length intervals case, we give a $2$-approximation sliding window algorithm with space $\tilde{\mathrm{O}}(|OPT|)$, and we show that any sliding window algorithm that computes a $(2-\varepsilon)$-approximation requires space $\Omega(L)$, for any $\varepsilon > 0$. - In the arbitrary-length case, we give a $(\frac{11}{3}+\varepsilon)$-approximation sliding window algorithm with space $\tilde{\mathrm{O}}(|OPT|)$, for any constant $\varepsilon > 0$, which constitutes our main result. We also show that space $\Omega(L)$ is needed for algorithms that compute a $(2.5-\varepsilon)$-approximation, for any $\varepsilon > 0$. Our main technical contribution is an improvement over the smooth histogram technique, which consists of running independent copies of a traditional streaming algorithm with different start times. By employing the one-pass $2$-approximation streaming algorithm by Cabello and P\'{e}rez-Lantero [Theor. Comput. Sci. '17] for Interval Selection on arbitrary-length intervals as the underlying algorithm, the smooth histogram technique immediately yields a $(4+\varepsilon)$-approximation in this setting. Our improvement is obtained by forwarding the structure of the intervals identified in a run to the subsequent run, which constrains the shape of an optimal solution and allows us to target optimal intervals differently.

翻译：我们首次在（流式）滑动窗口计算模型中研究区间选择问题。在该问题中，算法接收一条可能无限的线上区间流，目标是在任意时刻对最近$L$个区间中可能的最大互不相交区间子集保持一个近似解，其中$L$为整数。我们给出以下结果：- 在单位长度区间情形下，我们给出一个空间复杂度为$\tilde{\mathrm{O}}(|OPT|)$的$2$近似滑动窗口算法，并证明对于任意$\varepsilon > 0$，任何计算$(2-\varepsilon)$近似解的滑动窗口算法都需要$\Omega(L)$空间。- 在任意长度区间情形下，对于任意常数$\varepsilon > 0$，我们给出一个空间复杂度为$\tilde{\mathrm{O}}(|OPT|)$的$(\frac{11}{3}+\varepsilon)$近似滑动窗口算法，这是我们的主要结果。我们还证明对于任意$\varepsilon > 0$，计算$(2.5-\varepsilon)$近似解的算法需要$\Omega(L)$空间。我们的主要技术贡献是对平滑直方图技术的改进，该技术通过以不同起始时间运行传统流算法的独立副本来实现。通过采用Cabello和Pérez-Lantero [Theor. Comput. Sci. '17]提出的针对任意长度区间选择问题的单遍$2$近似流算法作为底层算法，平滑直方图技术在此设置下可直接得到$(4+\varepsilon)$近似解。我们的改进通过将一次运行中识别的区间结构传递给后续运行来实现，这约束了最优解的形状，使我们能够以不同方式定位最优区间。